Knowing how to rearrange formulae is a skill that will stretch far beyond GCSE Maths. Mathematical formulae are the basis of everything from architecture to accounting. Working with them is a skill that you may well need to use for many years to come.
To start off, let’s focus on what we’re trying to do here and what we need to work out. Our task is to find the variable that makes the rest of the formula work.
For instance, let us consider Newton’s second law of motion. This famous equation states that the force an object is subject to, is equal to its mass multiplied by the acceleration. This fundamental physical law is summarised in the simple formula F = ma, where F is the force, m the mass and a is the acceleration. Note that F and a are in bold because they are both vectors.
Rearranging processes depend on the characteristics of the formula. In the case of Newton’s second law, if we don’t know the mass and want to find it out, then we need to make the subject. We then need to divide both sides by the acceleration . Mathematically, this is done in the following way:
However, several rearranging processes exist. Because these processes depend on specific formulae, let’s investigate how these techniques can be applied through some practical examples.
1. When the subject appears in a fraction
In some cases, the variable will be expressed fractionally. Let’s look at an example.
The following formula describes the relation between the distance d, the initial velocity u, the final velocity v and the time t.
Let’s say we want to make u our subject. How should we rearrange the formula?
- Firstly we need to multiply both sides by . By doing this, the t and the 2 on the right-hand side cancel out:
2. Then we need to subtract v from both sides and then reorder the formula:
2. When the subject of a formula is squared
In these cases, it needs to be extracted by applying the square root on both sides. The reason for this is that the square root inverts the square.
Let’s recall that the formula for the area of a circle is:
We want to make r the subject.
- Firstly, we need to divide both sides by Π:
2. By taking the square root on both sides we have:
3. Finally, we can simplify this to:
3. If the subject of our formula is “trapped” inside a square root
This is just a case of extracting it by squaring both sides. Let’s consider this formula:
We’re going to make k the subject.
1. Firstly we need to square both sides to eliminate the square root:
2. Secondly, as we did in a previous example, we need to multiply both sides by to isolate k:
4. If the subject appears twice
In these cases, we need to factorise. Let us consider the following formula:
We want to make s the subject.
1. The first step is to factorise s
2. Then, by dividing both sides by 2 – G, we obtain:
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I am an applied mathematician and qualified secondary teacher. I have done research in the field of nuclear energy and am currently studying toward a PhD with the University of Cranfield in aerospace materials.